An olefin polymer of an alkenyl hydrocarbon having 5 or more carbon atoms, wherein said olefin polymer is an amorphous polymer having a Mn of 300,000 and a Mw/Mn of 2.40 or less, and substantially not having a melting point, and a process for producing said olefin polymer which comprises polymerizing an alkenyl hydrocarbon having 5 or more carbon atoms with an olefin polymerization catalyst obtained by contacting: a specific transition metal compound(A), an organoaluminumoxy compound (B) soluble in an aromatic solvent and water (C).